# Poor Symbol choice - Principles of Mathematical Analysis by Rudin

I've just started working through Rudin's Principles of Mathematical analysis. In my opinion, the choice of symbols doesn't promote "rigour" and encourages us to make needless assumptions.

For example, when defining an ordered field, the fourth axiom for addition states:

To every x belonging to F, corresponds an element -x belonging to F such that x+(-x)=0

I feel that the usage of "-x" and "0" are misleading. We're not assuming that we're talking of numbers. While it's true that we can still pretend we're talking abstractly, the choice of commonly used algebraic symbols is like a trap - encouraging us to subconsciously associate this definition of a field with rational numbers or real numbers.

I would have liked it if instead of "-x", he had written say (x') and instead of "0", he had written θ and instead of "1", he had chosen something like "I". This way, it encourages us to think abstractly instead of using "numbers" as our subconscious base.

For example, I was trying to prove something using Rudin's notation and unconsciously made the mistake of substituting -x with (-1)x when it wasn't proved yet! I caught the error while reading through the proof the second time, but because I've been using this notation for the past 20 years, it came naturally.

If on the other hand instead of -x, it was x', I would never have made the leap of logic to assume that x' = (I')x. It's just not the same thing. Even symbols like ">" and "<" are misleading because it encourages us (once again) to think of numbers. While it's still possible to go through the proofs and texts abstractly, it just makes it more difficult. And I wonder why this is necessary.

My question is, do you think Rudin deliberately chose "number based" terminology in order to make us think about numbers, or was it a mistake on his part?

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I would call it a tradition. You can use stars and other exotic typsetting symbols instead of . and + and I think texts on abstract algebra (only) tend to do it. But again analysis uses the fields of C and R, so that the authors are 'trapped' with + and .

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It doesn't assume numbers. Any time you have a commutative operation with an identity, you write that operation as + and the identity as 0, and inverses as -x. For example if you have matrices you write the 0 matrix even though that matrix isn't a number!. If you have functions you write the 0 function even though that function isn't a number.